The upper bound in Theorem REF  does depend on the order \(\rho \)  of \(f\)  and converges to 2 when \(\rho \rightarrow \infty \) .
This suggests that \(\text{{\rm HD}}(\mathcal {I}(f)) =2\)  whenever \(f\)  has infinite order, and this is the reason why in this case the behavior of \(\mathcal {I}(f)\)  with respect
to Lebesgue measure has been studied rather than the Hausdorff dimension of \(\mathcal {I}(f)\)  (see [1]} and the discussion preceding it). However, our result, i.e. Theorem REF , shows that \(\text{{\rm HD}}(\mathcal {I}(f))\)  does not really depend on the order. It certainly depends on the distribution of the poles but is also influenced
by the coefficients \(b_j\) .
